Figure 1. A nanoscale junction driven out of equilibrium.

On length scales long compared to the mean free path for inelastic electron-phonon scattering (beyond region A) the electronic and ionic degrees of freedom are assumed to approach approximately thermal distributions parametrized by identical (position-dependent) temperatures (elevated above the equilibrium temperature, T₀, by an amount depending on the thermal path to “the environment”). In region B, the electronic distributions quasi-equilibrate into a local Fermi-Dirac thermal form, parametrized by a position-dependent electronic temperature, T_e, greater than that of the ionic degrees of freedom. In region C, close to the ballistic junction relative to the mean free path for inelastic electron-electron scattering, the electronic distributions are non-thermal, resembling a linear combination of Fermi-Dirac distributions each with some electronic temperature T_e and energetically offset from each other by the applied bias, eV. Transmission through the ballistic region is through channels of transmittances τ_i. Quantum corrections to the ballistic conductance can arise through quantum interference of trajectories (black and red) involving scattering off disorder in the diffusive electrodes within a coherence length of the ballistic region. The nonequilibrium electronic distribution in regions C and D can lead to nonequilibrium populations of local phonon modes.